Division by Zero: Using Semi-structured Complex Numbers to Find the Inverse of a Singular Matrix and Show Its Applications

1. Introduction

1.1. Matrices and Their Determinants

Matrices are the ordered rectangular array of numbers, which are used to express linear equations. A matrix can be used to perform mathematical operations such as addition, subtraction and multiplication. A matrix consisting of $m$ rows and $n$ columns is called a $m\times n$ matrix. Matrices and their inverse allow accurate calculations to be produced quickly and compactly, which can be used to better control a mechanical device, industrial process or any process that can be presented as a system of equations.

Inverse of a matrix is defined usually for square matrices. For every $m\times m$ square matrix $A$, there exists an inverse matrix $A^{-1}$ such that:

$$A^{-1}=\frac{1}{\left|A\right|}\times adj\left(A\right)$$

$$A\times A^{-1}=I$$

where

$\left|A\right|$	Determinant of A
$adj\left(A\right)$	Adjoint matrix of A
$I$	Identity matrix

The inverse of a matrix depends on the determinant of the matrix. Important properties of the determinant of a matrix are given in Appendix A. According to standard definition, the determinant of a matrix should not be equal to zero. If the determinant is equal to zero, then this would mean that the matrix has no inverse because finding the inverse would involve division by zero. Matrices with determinant equal to zero are called singular matrices because they do not have inverses. Singular matrices have several disadvantages. These are listed in Table 1.

It is clear from Table 1 that if the inverse of a singular matrix can be found this would resolve a large number of problems in mathematics, science, engineering and any mathematically based subject where matrices are used.

Usually, when investigating potentially new insights into matrices $2\times 2$ matrices are used. This is because (1) they are one of the simplest types of matrices where matrix calculations can be done very easily and (2) finding the inverse of higher order square matrices comes down to finding the determinant of all the $2\times 2$ submatrices within it.

In this regard, when looking at a new method to find the inverse of a singular matrix, $2\times 2$ matrices can be investigated. However, additional tools are needed since finding the inverse of a singular matrix hinges on the ability to solve division by zero and the ability to provide an adequate geometric interpretation to this inverse.

Fortunately, a new number set called semi-structured complex numbers was recently created to solve the problem of division by zero in algebraic equations. Moreover, if a proper geometric interpretation is to be given geometries beyond standard Euclidean geometry needs to be considered. One such geometry that was found to be useful is projective geometry. It is instructive to understand what these tools are and how they are used to find and provide an interpretation to the inverse of a singular matrix.

1.2. Semi-Structured Complex Numbers: Enabling Division by Zero

Recently there has been a range of research involving division by zero. The problem of division by zero can simply be stated as: What is $\frac{a}{0}$ where “a” is any complex number. Table A1, Appendix B, shows sample research conducted on “division by zero”.

There have been several solutions to the problem the most recent being the invention of the semi-structured complex number set $\mathbb{H}$ [1]. The first attempt at creating this number set was riddled with issues [1], however, a second paper [2], written to reformulate and strengthen the theory of semi-structured complex numbers, produced several grounded and profound results. Table A2 in Appendix C, shows the major results developed in paper [2].

According to the first three results shown in Table A2 (Appendix C), Semi-structured complex number set can be defined as follows:

A semi-structured complex number is a three-dimensional number of the general form h = x + yi + zp; that is a linear combination of real (1), imaginary (i) and unstructured (p) units whose coefficients, x, y, z are real numbers.

The number $h$ is called semi-structured complex because it contains a structured complex part $\left(x+yi\right)$ and an unstructured component $\left(zp\right)$. The unstructured unit $p$ was redefined as:

$$p^n=\frac{\sqrt{2}\times \mathit{cos}\left(\frac{\pi }{2}n-\frac{\pi }{4}\right)}{f^n\left(1\right)}$$

(1)

where $f^n\left(c\right)$ is a composite function such that $f\left(c\right)=1-c$. Integer powers of $p$ yield the following cyclic results:

$$p^1=\frac{1}{0}, p^2=-1, p^3=-p, p^4=1, p^5=\frac{1}{0}, p^6=-1, p^7=-p \dots$$

(2)

$p$ does not belong to the set of complex numbers $\mathbb{C}$ (that is, $p\notin \mathbb{C}$), but belongs to a higher order number set $\mathbb{H}$ called the set of semi-structured complex numbers such that the set of complex numbers is a subset of $\mathbb{H}$ (that is, $\mathbb{C}\subset \mathbb{H}$).

Clearly from Table A2 (Appendix C), very little work has been done on the inverse of singular matrices. Moreover, very little has been done to show what the interpretation of such a result would be and where such a result would be useful.

1.3. Projective Geometry

Projective geometry is the study of geometric properties that remain unchanged by a projective transformation. A projective transformation is one that occurs when: points on one line are projected onto another line; points in a plane are projected onto another plane; or points in space are projected onto a plane.

Projections can be parallel or central. For example, the sun shining behind a person projects his or her shadow onto the ground. Since the sun’s rays are for all practical purposes parallel, it is a parallel projection.

A slide projector projects a picture onto a screen. Since the rays of light pass through a lens and the lens acts like a point through which all the rays pass, the projection is called a central projection (that is, the lens is the centre of the projection).

Some properties that remain unchanged by a projection are collinearity, intersection, and order. If three points are collinear, then when projected they will remain collinear. If two lines intersect then when projected their projections will remain intersected. If an object exists between two others, then their order will remain the same when they are projected onto a plane. Properties that can change during a projection are size and angles.

An ordinary Euclidean plane in which points are addressed with Cartesian coordinates, $(x,y)$, this plane can be converted to a projective plane by adding a line at infinity. The original visualization for this line at infinity is shown in Figure 1.

However, the visualization shown in Figure 1 was difficult to translate into mathematics and so a new way to visualize projective geometry was developed and is shown in Figure 2.

The Cartesian plane is represented as the plane $z=1$ and the “line at infinity” is now represented as a plane $z=0$. Hence a point on the Cartesian plane is give as $(x,y,1)$ or simply as $(x,y)$ and “points at infinity” have coordinates $(x,y,0)$ or simply as $\left(\frac{x}{0},\frac{y}{0}\right)$ (in semi-structured complex form $(xp,yp)$ since $\frac{1}{0}=p$).

The three-coordinate system $(x,y,z)$ is called the homogeneous coordinate system whilst the two-coordinate system $(x,y)$ is called Cartesian coordinate system. Both coordinate systems are essential in dealing with projective geometry calculations.

If a point has homogeneous coordinates $(x_1, x_2,x_3)$, then these homogeneous coordinates can be converted to Cartesian coordinates $(x,y)$ where $x=\frac{x_1}{x_3}$ and $y=\frac{x_2}{x_3}$. This enables Cartesian coordinates to be converted to homogeneous coordinates and vice-versa. For example, if a point has homogeneous coordinates $(7, 3, 5)$, these coordinates can be converted to Cartesian coordinates $\left(\frac{7}{5}, \frac{3}{5}\right)$. Conversely if a point has Cartesian coordinates $(4, 1)$, these coordinates can be converted to homogeneous coordinates $(4, 1, 1)$(or any multiple, such as $(12, 3, 3)$ or $(8, 2, 2)$).

The motivation for having homogeneous coordinates is very practical. Homogeneous coordinates are extensively used in computer graphics for computing transformations such as projection of a 3D scene (consisting of a set of 3D coordinates $(x_1, x_2,x_3)$) onto a viewing plane (with a set of coordinates $(x,y)$ such as a computer display).

Projective geometry theory can be applied to solving linear equations. For example, in the Euclidean plane $3x-y+4=0$ is the equation of a line. Written with homogeneous coordinates the equation of the line becomes $3\frac{x_1}{x_3}-\frac{x_2}{x_3}+4=0$. If this equation is multiplied through by $x_3$, the equation becomes $3x_1-x_2+4x_3=0$. The point $(1, 7)$ satisfied the original equation; the point $(1, 7, 1)$ satisfies the homogeneous equation. So, do $(0, 4)$ and $(\mathrm{0,4},1)$ and so on.

As another example, in the Euclidean plane, the lines $3x-y+4=0$ and $3x-y+10=0$ are parallel and have no point in common. In homogeneous coordinates, they do. In homogeneous coordinates the system $3x_1-x_2+4x_3=0$ and $3x_1-x_2+10x_3=0$ does have a solution. It is $(1, 3,0)$ or any multiple of $(1, 3,0)$. Since the third coordinate is zero, this is a point at infinity. In the Euclidean plane, the lines are parallel and do not intersect. In the projective plane, they intersect at infinity.

For the purposes of this paper, Cartesian coordinates and homogeneous coordinates are considered coordinates in projective space, whilst Euclidean coordinates are considered coordinates in Euclidean space. Note, however, that Cartesian coordinates and Euclidian coordinates are numerically the same.

Using these basic concepts of projective geometry and the semi-structured complex number set, the inverse of a singular matrix can be found and its geometric interpretation given.

1.4. Major Contributions

Given the potential importance of the inverse of singular matrices, the aim of this paper was:

Use semi-structured complex numbers and projective geometry to find the inverse of a general singular 2 × 2 matrix, give its geometric interpretation and its potential applications.

In the process of fulfilling the stated aim, this paper makes four major contributions:

• The inverse of a 2 ×2 matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is given by the equation:

  $$A^{-1}=\frac{p}{2ad}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$$

  (3)

  where $p$ is the unstructured unit.
• Singular matrices and their inverses can be used to find a unique solution to a pair of simultaneous equations that appear to have infinitely many solutions and a pair of simultaneous equations that appear to have no solution. This unique solution is called the principle inverse solution.
• Singular matrices map coordinates in projective space to coordinates in Euclidean space and their inverse maps coordinates in Euclidean space to coordinates in projective space.
• The inverse of a singular matrix proves that parallel lines in Euclidean space do not intersect but in projective space intersect at a “point at infinity”.
• The inverse of a singular matrix proves that collinear lines that intersect at infinitely many points in Euclidean space only intersect at one point in projective space.

The rest of this paper is devoted to showing how achieving the main aim of the paper led to these major contributions.

2. Inverse of a Singular Matrix

The inverse of a $2\times 2$ matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is given by the equation:

$$A^{-1}=\frac{p}{2ad}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$$

(4)

where $p$ is the unstructured unit.

Proof of Equation (4) is given in Appendix D. Essentially the inverse of a singular matrix consists of the adjoint matrix $adj\left(A\right)=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$, the determinant $\frac{1}{\left|A\right|}=\frac{p}{2ad}$.

3. Solving Simultaneous Equations Represented by a Singular Matrix

3.1. Solving Simultaneous Equations Which Appear to Have Infinitely Many Solutions

Suppose there is a pair of simultaneous equations with the form given by Equation (5). Here $m$ is a scalar.

$$\begin{array}{c}ax+by=f\\ max+mby=mf\end{array}$$

(5)

In terms of linear algebra Equations (5) represents the general equations of two collinear lines. The solution to these equations is the point of intersection of these two lines. Recall that collinear lines intersect at an infinitely many points in ordinary Euclidean space since the lines essentially rest on top of each other. This implies that simultaneous Equations (5) have infinitely many solutions. So, any method used to solve these equations must resolve this fact.

Suppose the matrix method is used to solve these equations. The matrix method for solving simultaneous equations involves converting the system of equations into the matrix form $AX=B$ (where $A$ is the matrix of coefficients, $X$ the matrix of variables and $B$ the matrix of constants) and then solving for $X$ using the expression $X=A^{-1}B$ (where $A^{-1}$ is the inverse of the matrix $A$).

Using this idea, Equations (5) can be represented in matrix form as shown in Equation (6).

$$\left(\begin{array}{cc}a& b\\ ma& mb\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}f\\ mf\end{array}\right)$$

(6)

Here $A=\left(\begin{array}{cc}a& b\\ ma& mb\end{array}\right)$ is a singular coefficient matrix, $X=\left(\begin{array}{c}x\\ y\end{array}\right)$ is the variable matrix and $B=\left(\begin{array}{c}f\\ mf\end{array}\right)$ is the matrix of constants. This system of equations that appear to have infinitely many solutions (according to standard algebra). However, using the inverse of a $2\times 2$ singular matrix given by Equation (4), a unique solution can be found for this system of equations. This solution is given by Equation (7).

$$\left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(mab\right)}\left(\begin{array}{cc}mb& -b\\ -ma& a\end{array}\right)\left(\begin{array}{c}f\\ mf\end{array}\right)$$

(7)

where $A^{-1}=\frac{p}{2\left(mab\right)}\left(\begin{array}{cc}mb& -b\\ -ma& a\end{array}\right)$.

Hence solving for the Equations given by (5) gives:

$$\begin{gathered} \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(mab\right)}\left(\begin{array}{cc}mb& -b\\ -ma& a\end{array}\right)\left(\begin{array}{c}f\\ mf\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(mab\right)}\left(\begin{array}{c}mbf-mbf\\ -maf+maf\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(mab\right)}\left(\begin{array}{c}mbf(1-1)\\ maf(-1+1)\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(mab\right)}\left(\begin{array}{c}mbf\left(0\right)\\ maf\left(0\right)\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{p}{2\left(mab\right)}\times mbf\left(0\right)\\ \frac{p}{2\left(mab\right)}\times maf\left(0\right)\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{mbf\left(0p\right)}{2\left(mab\right)}\\ \frac{maf\left(0p\right)}{2\left(mab\right)}\end{array}\right) \end{gathered}$$

According to semi-structured complex numbers $0p=1$. Hence:

$$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{f}{\left(2a\right)}\\ \frac{f}{\left(2b\right)}\end{array}\right)$$

(8)

Result (8) if substituted into Equations (5) yields the correct output. Hence, the inverse of a singular matrix was used to find a unique solution to a pair of simultaneous equations that represent collinear lines. The unique solution given by Result (8) is called the principle inverse solution.

The principle inverse solution found from Result (8) represents Cartesian coordinates in projective space. These coordinates can be converted to homogeneous coordinates $\left(x_1,x_2,x_3\right)$ in projective space using the conversion $(x,y)=(\frac{x_1}{x_3}, \frac{x_2}{x_3})$. Hence:

$$\begin{gathered} \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{f}{\left(2a\right)}\\ \frac{f}{\left(2b\right)}\end{array}\right)=\left(\begin{array}{c}\frac{x_1}{x_3}\\ \frac{x_2}{x_3}\end{array}\right) \\ \mathrm{This} \mathrm{implies} \mathrm{that} x_1=\frac{f}{\left(2a\right)} , x_2=\frac{f}{\left(2b\right)} \mathrm{and} x_3=1. \end{gathered}$$

Therefore, the solution to Equations (5) is the Cartesian coordinates $(\frac{f}{\left(2a\right)}, \frac{f}{\left(2b\right)})$ or the homogeneous coordinates $(\frac{f}{\left(2a\right)},\frac{f}{\left(2b\right)}, 1)$ in projective space. Substituting the homogeneous coordinates into Equation (5) yields Result (9) (the correct output).

$$\begin{array}{c}a\left(\frac{f}{2a}\right)+b\left(\frac{f}{2b}\right)=\left(1\right)f\\ ma\left(\frac{f}{2a}\right)+mb\left(\frac{f}{2b}\right)=\left(1\right)f\end{array}$$

(9)

A numerical example of solving simultaneous equations which appear to have infinitely many solutions is given in Appendix E PART 1.

3.2. Solving Simultaneous Equations Which Appear to Have No Solutions

Suppose there is a pair of simultaneous equations with the form given by Equation (10).

$$\begin{array}{c}ax+by=f_1\\ ax+by=f_2\end{array}$$

(10)

In terms of linear algebra Equations (10) represents the general equations of two parallel lines. The solution to these equations is the point of intersection of these two lines. Recall that parallel lines do not intersect in ordinary Euclidean space. However, they do intersect at a “point at infinity” in projective space. So, any method used to solve these equations must demonstrate these two facts.

Suppose the matrix method is used to solve these equations. The matrix method for solving simultaneous equations involves converting the system of equations in the matrix form $AX=B$ (where $A$ is the matrix of coefficients, $X$ the matrix of variables and $B$ the matrix of constants) and then solving for $X$ using the expression $X=A^{-1}B$ (where $A^{-1}$ is the inverse of the matrix $A$).

Using this idea, Equations (10) can be represented in matrix form as shown in Equation (11).

$$\left(\begin{array}{cc}a& b\\ a& b\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)$$

(11)

Here $A=\left(\begin{array}{cc}a& b\\ a& b\end{array}\right)$ is a singular coefficient matrix, $X=\left(\begin{array}{c}x\\ y\end{array}\right)$ is the variable matrix and $B=\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)$ is the matrix of constants. This system of equations that appear to have no solution (according to standard algebra). However, using the inverse of a $2\times 2$ matrix given by Equation (4), these systems of equations has a unique solution given by Equation (12).

$$\left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(ab\right)}\left(\begin{array}{cc}b& -b\\ -a& a\end{array}\right)\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)$$

(12)

where $A^{-1}=\frac{p}{2\left(mab\right)}\left(\begin{array}{cc}b& -b\\ -a& a\end{array}\right)$. Hence solving for Equation (11) gives:

$$\begin{gathered} \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(ab\right)}\left(\begin{array}{cc}b& -b\\ -a& a\end{array}\right)\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(ab\right)}\left(\begin{array}{c}b{f}_1-b{f}_2\\ -a{f}_1+a{f}_2\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\frac{p}{2\left(ab\right)}\left(\begin{array}{c}b\left(f_1-f_2\right)\\ a\left(-f_1+f_2\right)\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{p}{2\left(ab\right)}\times b\left(f_1-f_2\right)\\ \frac{p}{2\left(ab\right)}\times a\left(f_2-f_1\right)\end{array}\right) \\ \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{p\left(f_1-f_2\right)}{2a}\\ \frac{p\left(f_2-f_1\right)}{2b}\end{array}\right) \end{gathered}$$

Hence:

$$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{\left(f_1-f_2\right)}{2a}p\\ \frac{\left(f_2-f_1\right)}{2b}p\end{array}\right)$$

(13)

The result shown in Result (13) if substituted into Equation (10) yields the correct output. Hence, the inverse of a singular matrix was used to find a unique solution to a pair of simultaneous equations that represent parallel lines. The unique solution given by Result (13) is called the principle inverse solution.

The principle inverse solution found from Result (13) represents the Cartesian coordinates in Euclidean space (governed by Euclidean geometry). These coordinates can be converted to homogeneous coordinates in projective space (governed by projective geometry) coordinates using the conversion $(x,y)=(\frac{x_1}{x_3}, \frac{x_2}{x_3})$ where $\left(x_1,x_2,x_3\right)$ are homogeneous coordinates in the projective space. Hence:

$$\begin{gathered} \left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}\frac{\left(f_1-f_2\right)}{2a}p\\ \frac{\left(f_2-f_1\right)}{2b}p\end{array}\right)=\left(\begin{array}{c}\frac{\left(f_1-f_2\right)}{\left(2a\right)\left(0\right)}\\ \frac{\left(f_2-f_1\right)}{\left(2b\right)\left(0\right)}\end{array}\right)=\left(\begin{array}{c}\frac{x_1}{x_3}\\ \frac{x_2}{x_3}\end{array}\right) \\ \mathrm{This} \mathrm{implies} \mathrm{that} x_1=\frac{\left(f_1-f_2\right)}{2a} , x_2=\frac{\left(f_2-f_1\right)}{2b} \mathrm{and} x_3=0. \end{gathered}$$

Therefore, the solution to Equations (5) is the Cartesian coordinates $(\frac{\left(f_1-f_2\right)}{2a}p, \frac{\left(f_2-f_1\right)}{2b}p)$ or the homogeneous coordinates $(\frac{\left(f_1-f_2\right)}{2a}, \frac{\left(f_2-f_1\right)}{2b},0)$ in projective space. Substituting the homogeneous coordinates into Equation (10) yields Result (14) (the correct output).

$$\begin{array}{c}a\left(\frac{\left(f_1-f_2\right)}{2a}\right)+b\left(\frac{\left(f_2-f_1\right)}{2b}\right)=\left(0\right)f_1\\ a\left(\frac{\left(f_1-f_2\right)}{2a}\right)+b\left(\frac{\left(f_2-f_1\right)}{2b}\right)=\left(0\right)f_2\end{array}$$

(14)

Any point in projective space that has homogeneous coordinate $x_3=0$ represents a “point at infinity” in projective space. This implies that solution of the equation using the inverse of a singular matrix lead to the fact that parallel lines intersect at a “point at infinity” in projective space. This agrees with currently established mathematics of projective geometry. A numerical example of solving simultaneous equations which appear to have no solutions is given in Appendix E PART 2.

Based on the examples given in Section 3.1 and Section 3.2 it is clear that there is a relationship between Euclidean space and coordinates in projective space. The simultaneous equations shown in Equations (5) and Equations (10) are collinear and parallel lines respectively in Euclidean space. These equations when put in matrix form result in singular matrices. The solution to these equations (found by using the inverse singular matrix) results in points that rest in projective space. This implies that singular matrices and their inverses results in transformations move objectives from Euclidean space to projective space and vice versa. Therefore, singular matrices and their inverse establish a strong relationship between these two geometries.

4. Discussion

It is clear from Section 3 that semi-structured complex numbers can be used to find the inverse of a singular matrix and this inverse can be used to find unique solutions to a pair of simultaneous equations that appear to have infinitely many solutions and a pair of simultaneous equations that appear to have no solution. The unique solution is called the principle inverse solution.

Projective geometry was used to clarify the meaning of the principle inverse solutions. Using projective geometry, it was clear that singular matrices map coordinates in projective space to coordinates in Euclidean space and the inverse of a singular matrix maps coordinates in Euclidean space to coordinates in projective space.

It is also important to note that the inverse of a singular matrix proves that parallel lines in Euclidean space do intersect at a “point at infinity” in projective space. Additionally, collinear lines that intersect at an infinitely many points in Euclidean space only intersect at one point in projective space.

These results are useful in producing time and cost savings in processes that are analysed using matrices and often require extensive workarounds when singular matrices are encountered. Additionally, the fact that the inverse of a singular matrix can be found implies that these matrices can now be used in areas of mathematics such as cryptography, graph theory and any other area of science and engineering where singular matrices may pose a problem.

5. Conclusion

Matrices and their inverses allow accurate calculations to be produced quickly and compactly, which can be used to better control any process that is represented by a system of equations. In cases where singular matrices are encountered, this may require extensive workaround to solve these systems of equations. This workaround often cost time and money.

Singular matrices have no inverse because finding their inverse involves division by zero. However, recently a new number set called semi-structured complex numbers has been developed to enable division by zero in regular algebraic equations. The aim of this research is to demonstrate that the inverse of a singular matrix can be found using semi-structured complex numbers.

The results reveal (1) Singular matrices and their inverses can be used to find a unique solution to a pair of simultaneous equations that appear to have infinitely many solutions and a pair of simultaneous equations that appear to have no solution; (2) Singular matrices map coordinates in projective space to coordinates in Euclidean space and their inverse maps coordinates in Euclidean space to coordinates in projective space; (3) the inverse of a singular matrix proves that parallel lines in Euclidean space do intersect but in projective space intersect at a “point at infinity” and collinear lines that intersect at infinitely many points in Euclidean space only intersect at one point in projective space.

These results provide a firm foundation for the use of semi-structured complex numbers in mathematics.